of multiple statistical populations; each fracture plane is an independent ... populations is used to simulate groundwater flow in the fractured medium. An.
Stochastic Environmental Research and Risk Assessment 16 (2002) 155–174 Ó Springer-Verlag 2002 DOI 10.1007/s00477-002-0091-7
3D inverse modelling of groundwater flow at a fractured site using a stochastic continuum model with multiple statistical populations H. J. W. M. Hendricks Franssen, J. J. Go´mez-Herna´ndez 155 Abstract. 3D groundwater flow at the fractured site of Aspo¨ (Sweden) is simulated. The aim was to characterise the site as adequately as possible and to provide measures on the uncertainty of the estimates. A stochastic continuum model is used to simulate both groundwater flow in the major fracture planes and in the background. However, the positions of the major fracture planes are deterministically incorporated in the model and the statistical distribution of the hydraulic conductivity is modelled by the concept of multiple statistical populations; each fracture plane is an independent statistical population. Multiple equally likely realisations are built that are conditioned to geological information on the positions of the major fracture planes, hydraulic conductivity data, steady state head data and head responses to six different interference tests. The experimental information could be reproduced closely. The results of the conditioning are analysed in terms of ensemble averaged average fracture plane conductivities, the ensemble variance of average fracture plane conductivities and the statistical distribution of the hydraulic conductivity in the fracture planes. These results are evaluated after each conditioning stage. It is found that conditioning to hydraulic head data results in an increase of the hydraulic conductivity variance while the statistical distribution of log hydraulic conductivity, initially Gaussian, becomes more skewed for many of the fracture planes in most of the realisations.
1 Introduction One of the main problems we have to face if we want to simulate groundwater flow and mass transport is the relative scarcity of experimental information. A limited amount of measurement data is available as compared to the data needs of
H. J. W. M. Hendricks Franssen (&), J. J. Go´mez-Herna´ndez Department of Hydraulic and Environmental Engineering, Technical University of Valencia, C./ Camino de Vera s/n, 46071 Valencia, Spain Thanks are due to SKB, the Swedish Nuclear Fuel and Waste Management Company, for supplying the experimental data in the context of the TRUE Block Scale project. The financial support of ENRESA, the Spanish Nuclear Waste Management Company is gratefully acknowledged.
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the simulation models. The scarcity of data makes it even more important to use the available data in the most efficient way and to estimate the uncertainty associated with the model predictions. It is in this context that statistically based methodologies were developed for the solution of the inverse groundwater flow problem (e.g. Kitanidis and Vomvoris, 1983; Hoeksema and Kitanidis, 1984; Dagan, 1985; Carrera and Neuman, 1986a, b; Rubin and Dagan, 1987a, b; Gutjahr and Wilson, 1989; Sahuquillo et al., 1992; Gutjahr et al., 1994; RamaRao et al., 1995; Oliver et al., 1997; Hanna and Yeh, 1998; Hu, 2000). However, in few occasions these methodologies were applied to strongly heterogeneous media in real case scenarios. The sequential self-calibrated approach (Sahuquillo et al., 1992; Go´ mezHerna´ ndez et al., 1997; Capilla et al., 1997) is not limited to mildly or moderately heterogeneous formations and has been successfully applied on strongly heterogeneous media (e.g. Capilla et al., 1998). The method was also applied on the modelling of 3-D groundwater flow in a fractured medium (Hendricks Franssen et al., 1999b; Go´ mez-Herna´ ndez et al., 2000). This paper presents the application of the sequential self-calibrating method on another 3D fractured site, treats the fracture plane positions as deterministically instead of stochastically and incorporates transient head data from multiple transient experiments. A stochastic continuum (SC) approach with multiple statistical populations is used to simulate groundwater flow in the fractured medium. An SC approach was chosen instead of a discrete fracture approach for the following reasons: (i) heterogeneity on transmissivity within the fracture planes, and within the background fracturing is naturally included in the model, (ii) flow at fracture intersections does not require any special modelling, and (iii) the model is easy to condition to hydraulic conductivity data, and as will be shown, also to the hydraulic head data. It was not necessary to consider turbulent flow; only in a few hydraulic tests with very large pumping rates, it appeared that the flow might be turbulent near the pumping sections. Otherwise, given the fracture widths and the flow rates, there is no justification to consider turbulent flow. The main purpose of this paper is to illustrate the application of the methodology on a complex real-world case study in a fractured rock site. The study zone is located at the Aspo¨ underground rock laboratory, in the Aspo¨ island near the village of Oskarshamn. Oskarshamn is situated on the coast of South Sweden, about 300 km south of Stockholm. At the Aspo¨ underground rock laboratory experiments are carried out related with the underground storage of nuclear waste. This study was carried out in the context of the TRUE (Tracer Retention Understanding Experiment) Block Scale Project, in which several international research groups participated. One of the principal aims of the project was the prediction of the transport of contaminants in a network of fractures on the scale of tens of metres. This paper presents the results for the last simulation round, in which the largest amount of experimental information was available. However, the inverse modelling was also carried out for earlier simulation rounds, in which less experimental information was available. The paper compares briefly the results for these earlier simulation rounds with the last simulation round. This paper is organized as follows. First the methodology is described. Then, the case study is presented in which the implementation of the structural model, the hydraulic conductivity data and the piezometric head data are highlighted in separate sections. The simulation outcomes are analysed and finally some discussion and conclusions are presented.
2 Methodology The sequential-self calibrated approach (Sahuquillo et al., 1992; Go´ mez-Herna´ ndez et al., 1997; Capilla et al., 1997) is a methodology for the stochastic inverse modelling of groundwater flow. A sufficiently large number of equally likely realisations, all of them conditional to conductivity data and piezometric head data, is generated. Details are given in the above mentioned papers. Below a short summary of the main steps is given; the procedure outlined below is implemented in the software INVERTO (Hendricks Franssen, 2001). (1) Hydraulic conductivity seed realisations are generated conditional to the hydraulic conductivity data by MultiGaussian or non-MultiGaussian sequential simulation (Go´ mez-Herna´ ndez and Srivastava, 1990; Go´ mez-Herna´ ndez and Journel, 1993). The study domain may correspond to multiple populations, i.e., different lithological facies or rock types. In that case, the hydraulic conductivity values are generated for each population conditioned to the data on that population. In the present study each of the 21 major fracture planes is treated as a different population. For each population a variogram has to be defined. The statistical populations are not cross-correlated. If a grid cell belongs to two populations, the simulated hydraulic conductivity at that grid cell is conditioned to measurements taken in both zones. This is for example the case for grid cells that are intersected by two fracture planes. Equally likely realisations of storativity can be generated also, along with the hydraulic conductivity realisations if deemed necessary (Hendricks Franssen et al., 1999a). Storativity and conductivity may be cross-correlated, in that case co-simulation has to be used. In the present study we did not consider heterogeneous storativity, if we had we should take care in the analysis of the possible cross-correlation and its incorporation in the simulation. (2) The steady-state and/or transient groundwater flow equations are solved for the hydraulic conductivity–storativity realisation couples, with given external stresses, boundary and initial conditions. The solution is obtained using a sevenpoint block centred finite differences approach using the harmonic (for 1-D studies) or the geometric mean (2-D or 3-D studies) to calculate the interblock conductivities. (3) An objective function J is computed as the sum of squared differences between measured and simulated heads. The aim is to minimise the objective function; the optimisation parameters are the hydraulic conductivities (and possibly also the storativity and the prescribed heads along the boundaries) at a limited number of selected locations, the so-called master blocks (see step 4). If J is smaller than a user-defined tolerance value the realisation is considered conditional to the experimental head data and the conditioning stops. The simulation also stops if a maximum number of iterations is reached or in the case that after too many iterations the value of J does not decrease. Otherwise, the algorithm enters into a non-linear optimisation algorithm that minimizes J. (4) The objective function is parameterised as a function of the perturbations of hydraulic conductivity (and possibly storativity and prescribed heads at the boundaries) at a number of selected locations (so-called master blocks). However, the master blocks are used as a way to parameterise the entire realisation, and this justifies the location of the master blocks on a regular grid. There should be as many master blocks as to have enough degrees of freedom so that the optimisation yields a solution and as little as possible to make the optimisation computationally feasible and stable. There is no problem if we overparameterise (more degrees of freedom than observation points to match) the optimisation
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problem; it is not a classical optimisation problem seeking a unique solution; we seek many equally likely solutions all of which are consistent with the experimental information (see also Capilla et al., 1997). The perturbations elsewhere within the realisations are obtained by linear interpolation of the perturbation at the master blocks. Hendricks Franssen et al. (1999a) detail the non-linear optimisation procedure, in which the adjoint-state formulation is used to determine the gradient of the objective function. In the context of the study presented in this paper, and since several populations are considered, master blocks are located within each population, and the perturbation of the master blocks spread only to the blocks within each population. (5) Once the optimal perturbations are found, they are added to the seed fields and new seed fields are generated. As stated before, a porous medium equivalent is used to model the site. The grid cells in which the study domain is discretized are classified according to the fracture plane equations, all cells intersected by a given fractured plane belong to a separate population with its own statistical properties. (Cells intersected by several fracture planes are assigned the geometric average of the cell values obtained as if they belong to each fracture individually.) As an aside, we would like to mention that, although, in this case the fractures are given deterministically, the algorithm was initially implemented to be used in conjunction with a stochastic facies generator that allows treating stochastically the positions of the fractures (Hendricks Franssen et al., 1999b; Go´ mez-Herna´ ndez et al., 2000).
3 Site description and experimental information The underground rock laboratory at Aspo¨ has been mined in a crystalline rock formation. A spiralling tunnel is the main access to the different niches from which the experiments are performed. The tunnel is acting as an artificial sink. For the experiment concerning this study a number of boreholes has been drilled from the rock into the formation at a depth ranging between 300 and 550 m below ground surface. These boreholes have served to conduct mineralogical analysis, hydraulic tests and tracer tests. The experimental information has been supplied to us by SKB, the Swedish Nuclear Fuel and Waste Management Company, in the context of the TRUE Block Scale project. The experimental data are not yet available in publically available reports. The geological information obtained from the boreholes, mostly concerning major fracture intersections, served to build a structural model of the fracture plane positions. The structural model evolved with the drilling of each new borehole. The current structural model, referred to as the March 99 structural model, is deemed sufficiently accurate and will not be subject of further updating. In total 21 fracture planes are incorporated in the model. The fracture planes are subvertical and NNW-SSE oriented. The hydraulic conductivities, for both the major fracture planes and the background, were determined using various types of experiments: flow logging on 5 m intervals, build-up tests and flow difference measurements. Hydraulic head was continuously monitored at the site in all boreholes and all interpacker intervals. After drilling the tunnel hydraulic head values have kept on decreasing very slowly from their initial values due to the influence of the tunnel. Given that this decrease is slow and that the time frame of our model predictions is small, we will assume that the measured hydraulic head values are at steadystate. Furthermore, a regional groundwater flow model was available that could be used to obtain prescribed head values along the model boundaries.
Finally, data from a number of hydraulic interference tests was available. These interference tests were of two types, short tests of about 30-min duration and longer tests of up to 72-h duration. Of all these tests, we used as conditioning data those tests, the target of which was a set of fractures that had special relevance for the tracer tests planned for the future. More specifically we used five shortduration tests, and one long-duration test.
4 Objective The final objective of this exercise is the hydraulic characterization of the block of rock surrounding the location in which a number of reactive transport tracer tests are planned. As intermediate goals we would test the self-calibrated algorithm on multiple populations, three dimensions, and conditioning not only to steady-state data but also to multiple transient events. If, in addition, multiple conditional realisations are generated, an idea about uncertainty at the site will be obtained. 5 Implementation of the structural model The groundwater flow model assumes that flow takes place in both the main fracture planes and the background fractures. Therefore the full domain is considered in the model and not only the main fracture planes. The elements not belonging to the main fractures are referred to hereafter as ‘‘background’’. The background cells contain both rock matrix and background fracturing. The flow model extends over an area of 247 m (1786.7–2033.3 m Easting) by 227 m (7046.7–7273.3 m Northing) by 287 m ()573.3 m to )286.7 m elevation). The area is divided into 37 by 34 by 43 cubic grid cells of size 20/3 m. It would have been preferable to use a smaller grid cell size, however, in order to reduce the required CPU time it was not possible to reduce the grid cell sizes more. The observed size of the fracture planes is in general less than 1 m, so that cubic grid cells of 6.6667 m are a very coarse representation of the fracture planes. Another consequence of the ‘‘coarse’’ fracture planes is the artificial enhancement of the connection between the different fracture planes near fracture intersections. The so-called March 99 structural model was used to classify the grid cells of the model. Figure 1 shows a binary classification of the model cells distinguishing between those cells that are intersected by one or more cells and those which are not. The model includes 21 fracture planes. 6 Simulation of hydraulic conductivity seeds A total of 270 hydraulic conductivity data values for the background, and 23 values for the fractures were available. Given the variety of conductivity estimates available, the following decision was taken: the fracture conductivities were determined from transient tests if available, otherwise from steady-state tests; if more than one transient test is available, the geometric mean of the estimates is retained. Figure 2 shows the histogram of the hydraulic conductivity data (some fracture planes had no data, and the maximum number of data values in a given fracture planes was four). The hydraulic conductivity data were used to generate the conditional hydraulic conductivity seeds by sequential Gaussian simulation (Go´ mez-Herna´ ndez and Journel, 1993). Each seed is obtained by merging the independent generation of each of the 22 populations (21 fracture planes plus background). Each population is generated in the log-space and later is backtransformed into decimal space. To
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Fig. 1. Classification of the model grid cells into cells corresponding to fracture planes (white) and cells not corresponding to any fracture plane (black)
Fig. 2. Histogram of the hydraulic conductivity data
generate each of the populations, besides the conditioning data, it is necessary to provide the overall mean conductivity and a variogram. The mean conductivity of each population is obtained from the data values of that population, except for those fracture planes with no data, in which case a mean equal to the mean of all fractures is used. For the variograms, there is no data in any population to estimate a meaningful variogram, therefore a variogram model is adopted. For all 22 populations, the same variogram for log with a 10-conductivity is used, spherical, 2 2 range of 40 m, a nugget 0.1 (log(m/s)) and a sill of 1.0 (log(m/s)) . From other, synthetic, studies it was found that the results are relatively robust to variogram mistakes (Capilla et al., 1997; Hendricks Franssen, 2001). When merging the 22 populations, fracture plane cells always prevail over background cells. However, those cells that are intersected by more than one
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Fig. 3. Hydraulic conductivity seed realisation. This realisation is conditioned to the information on the fracture plane positions and hydraulic conductivity data
fracture plane form the fracture intersection zones (FIZs). The conductivity assigned to a grid cell in a FIZ is equal to the geometric mean of the conductivities of each of the fractures intersecting the cell. Figure 3 shows a view of a hydraulic conductivity seed.
7 Conditioning to steady-state hydraulic head data The seed hydraulic conductivity fields are the starting point to obtain conductivity fields conditioned to hydraulic head data. Boundary conditions had to be supplied to the model. Along all the six sides of the cube prescribed heads were imposed. The prescribed heads were obtained from the regional groundwater flow model. Part of the tunnel, with atmospheric pressure, is also included in the model. Since it was known that the hydraulic heads were not completely at steadystate and kept decreasing by the sink effect of the tunnel, and since the selfcalibrated approach is capable to calibrate the boundary conditions too, it was decided to use the heads from the regional groundwater flow model as a starting value for the calibration of the boundary conditions, too. A total of 51 steady-state head data, from nine boreholes, were used as conditioning data. One thousand master blocks were used in the conditioning procedure; 100 master blocks are located in the background on a regular grid with a random starting point, and 900 master blocks are located randomly on the different fracture planes with a minimum of 12 master blocks for each fracture plane. For the calibration of the boundary conditions, 100 master blocks were considered, and a maximum deviation of 20 m from the heads coming from the regional groundwater flow model is allowed. The locations of the master blocks vary during each iteration of the non-linear optimisation process. During the inverse conditioning step, the possibility of treating the FIZs as an independent population arose, and after some tests it was decided to treat them so. The alternative to treat the FIZs as the geometric average of the conductivity values of the intersecting fractures was discarded after considering that at the intersections conductivity could be either enhanced or reduced depending on physical and mechanical factors.
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8 Conditioning to transient head data The conditioning to the transient test information is carried out on a smaller grid in order to reduce the needed CPU time. Even on the smaller grid it takes about one week of CPU time (on the fastest PC’s actually available) to condition a single realisation. Advantages are that more realisations could be conditioned to transient test information and also the numerical stability of the groundwater flow solution was improved. The smaller grid has 20 by 20 by 22 grid cells (8800 grid cells in total), with the same grid cell sizes as for the steady-state groundwater flow simulation, and extends from 1853.3 to 1986.7 m Easting, 7113.3 to 7246.7 m Northing and )573.3 to )426.7 m elevation. For the conditioning to transient data, we consider only drawdowns, not absolute piezometric heads, and since the transient tests to which the realisations are conditioned do not last enough for their effects to reach the boundary cells, zero drawdown initial conditions were used, and zero drawdown at the boundary cells prescribed. The seed fields used for the conditioning to transient head data are those obtained after conditioning to steady-state data. In total five short-term (about 30-min duration) tests and a long-term test (72-h duration) were used in the conditioning procedure. At 13 monitoring locations, the transient response to all tests was used as conditioning data; in addition, at 16 additional locations, the transient response to the long-term test was also available and used for conditioning. A decision was taken to calibrate a single homogeneous storativity value for all realisations, instead of considering spatially variable storativities. The main reason was to save CPU time during the calibration of the seed fields, although it would have been possible to calibrate jointly couples of realisations of conductivity and storativity. For the calibration of the single homogenous storativity the seed value was 10)6 m)1. A total of 1500 master blocks were used in the conditioning to transient head data, 500 of them located in the background and 1000 in the fracture planes. 9 Simulation results Conditioning to hydraulic conductivity and steady-state head data A total of 12 conductivity realisations were generated conditioned to hydraulic conductivity and steady-state head data. For each of the seed conductivity realisations, the average hydraulic conductivity in the background, each of the 21 fracture planes and the fracture intersection zones is calculated. These averages for each of the realisations are, in turn, averaged over the 12 realisations (see Table 1). The average background hydraulic conductivity over the 12 realisations is )10.0 log10 (m/s) and the average fracture plane conductivity )6.7 log10 (m/s), a difference of more than three orders of magnitude. The differences in average conductivity among the fracture planes are also considerable and the most conductive fracture plane (fracture plane #5; )4.7 log10 (m/s)) is more than three orders of magnitude more conductive than the less conductive fracture plane (fracture plane #13; )8.1 log10 (m/s)). The differences in average fracture plane conductivities among realisations are small. In the next step, the seeds were conditioned to steady-state head data. Figure 4 shows the reproduction of the steady-state head data. Figure 5 gives the results from one of the realisations with the updated hydraulic conductivities and the
Table 1. Average fracture plane conductivities after different conditioning stages. The averages are calculated over 12 realisations after conditioning to geology and conductivity data and after conditioning to steady-state head data. The averages are calculated over 10 realisations after conditioning to five short-term tests and over eight realisations after conditioning to long duration tests
Background Fr. Plane 1 Fr. Plane 2 Fr. Plane 3 Fr. Plane 4 Fr. Plane 5 Fr. Plane 6 Fr. Plane 7 Fr. Plane 8 Fr. Plane 10 Fr. Plane 12 Fr. Plane 13 Fr. Plane 15 Fr. Plane 16 Fr. Plane 17 Fr. Plane 18 Fr. Plane 19 Fr. Plane 20 Fr. Plane 21 Fr. Plane 22 Fr. Plane 23 Fr. Plane 24 FIZ
Conditioned to geology and K data
Conditioned to steady heads
Conditioned to five shortterm tests
Conditioned to long duration tests
)10.01 )6.59 )6.34 )6.54 )7.27 )4.71 )7.94 )5.58 )5.45 )7.01 )7.83 )8.05 )7.64 )5.36 )5.96 )7.04 )6.70 )6.95 )8.05 )7.22 )7.05 )6.34 )6.58
)10.21 )6.59 )6.11 )6.41 )7.13 )5.46 )8.33 )6.32 )5.35 )7.00 )7.88 )7.38 )7.78 )5.70 )5.97 )6.93 )6.93 )6.81 )7.61 )7.07 )7.06 )6.33 )6.46
)10.13 )6.61 )6.06 )6.18 )7.10 )5.38 )8.22 )6.12 )5.52 )7.04 )7.85 )7.29 )7.80 )5.78 )5.96 )6.96 )6.95 )6.65 )7.48 )7.01 )7.07 )6.33 )6.99
)10.21 )6.62 )6.17 )6.21 )7.15 )5.41 )8.27 )6.13 )5.69 )7.10 )7.86 )7.33 )7.80 )5.89 )6.02 )6.84 )6.92 )6.79 )7.80 )7.20 )7.04 )6.34 )7.57
Fig. 4. Comparison of the simulated and measured steady state head values, for a realisation
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Fig. 5a–j. Results after conditioning to steady state head data, for one single realisation. a 3-D steady state head solution, steady state head solution for slices at b 550 m depth, c 470 m depth, d 390 m depth and e 310 m depth. f An updated 3-D hydraulic conductivity image, slices of updated hydraulic conductivities at g 550 m depth, h 470 m depth, i 390 m depth and j 310 m depth
spatial distribution of steady-state heads after conditioning. From the figure it can be seen that the prescribed head boundaries force a flow in the direction towards the tunnel on the North and East sides. The conditioning to steady-state head data induces important local conductivity changes with regard to the seed fields (see Fig. 6). However, there is not a clear trend in these changes. The following observations can be made after analysing the results: Conditioning to steady-state head data hardly modifies the average conductivity contrast between background and fracture planes. Just a small increase is observed due to a decrease in the background conductivity (average over 12 realisations )10.2 log10 (m/s)) (see Table 1). The contrast on average conductivities among the fracture planes, decreases (see again Table 1). In particular, fracture plane #5 with the largest average conductivity in the seed realisations decreases from )4.7 log10 (m/s) to )5.5 log10
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Fig. 6a–e. Hydraulic conductivity perturbation during conditioning to steady-state head data, for one realisation. a 3-D image of hydraulic conductivity perturbation, hydraulic conductivity perturbation for slices at b 550 m depth, c 470 m depth, d 390 m depth and e 310 m depth
(m/s). On the contrary, fracture plane 13 with the smallest average conductivity in the seed realisations increases from )8.1 log10 (m/s) to )7.4 log10 (m/s). The largest difference between average conductivities after conditioning is between fracture plane 8 ()5.4 log10 (m/s)) and fracture plane 6 ()8.3 log10 (m/s)). The contrast in average fracture plane conductivities among realisations increased during the conditioning to steady-state head data. This indicates that the estimated prior ensemble variances were too low. Also related with a too low estimate of the prior ensemble variance is the increase in spatial variability of hydraulic conductivity within the realisations. Although the average fracture plane conductivities do not change much, the
variances of the fracture plane conductivities increase. This is specific for the dataset used in this study. More over, the distributions of the fracture plane conductivities become more skewed for many of the fracture planes in most of the twelve realisations.
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Together with the hydraulic conductivities also the prescribed heads along the boundaries were allowed to change during the conditioning. The most important head perturbations, with respect to the head values deduced from the regional groundwater flow model of 1996 are made on the Northern boundary of the model, near the tunnels. This may indicate that the influence of the tunnel on the head distribution is stronger than was supposed by the boundary heads prescribed initially. All the simulations showed, in general, a head decrease on the boundaries with a larger decrease in the zones near the tunnels on the Northern and Eastern model boundary. The observed decrease of the boundary heads in the calibration is consistent with the observed decreases of the ‘‘steady-state head’’ values between the years 1996 and 1999. Conditioning to transient head data Of the 12 realisations conditioned to steady-state head data, only eight could be conditioned to transient head data, that is, the optimisation algorithm was not capable to further calibrate the steady-state calibrated fields to obtain a solution close enough to the transient head information. If an input realisation is too far from reality, it is very difficult to converge to an optimal solution. For all eight realisations the reproduction of the experimental heads was satisfactory. Figure 7 shows, for a single realisation, the head drawdowns in the transient tests. It is clear from the figure that the head decreases extend along the orientation of the fracture planes, in the NNW–SSE direction and in the vertical direction. It was checked that the drawdowns induced by the tests did not reach the boundaries, so that the zero drawdown prescribed boundary conditions are consistent with the tests. In addition to conditioning the conductivity fields, for each realisation a single storativity value was calibrated. Table 2 gives the calibrated single storativity values for all the realisations; these values are very similar for all realisations. With regard to the changes in conductivity during the process of conditioning to the transient data, these modifications are very localized in each fracture plane and do not represent large changes neither for the background nor for the fracture planes. The average conductivities are hardly affected by Table 2. Calibrated single storativity values for the different realisations. Only eight realisations could be generated conditioned to transient data Simulation number
Calibrated storativity (m)1)
3 4 5 6 9 10 11 12
1.5010)8 1.9010)8 2.0710)8 2.2010)8 1.8010)8 2.2010)8 3.6010)8 1.3010)8
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Fig. 7a–e. Hydraulic head drawdown during the long-term pumping test. This solution corresponds to one realisation conditioned to all the pieces of information. a 530 m depth, b 510 m depth, c 490 m depth, d 470 m depth (pumping location) and e 450 m depth
the calibration, each fracture changes in average less than 0.3 log10 (m/s) units (see Table 1). The fracture planes that are the main target of the transient tests (fracture planes 6, 7, 13, 19, 20, 21 and 22, which are likely candidates for tracer tests in a latter stage of the experiment) decrease their average fracture plane conductivity up to a maximum value of 0.3 log10 (m/s) units. The most important conductivity changes are observed in the FIZs; the average
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conductivity for the FIZ grid cells decreases 0.6 log10 (m/s) units. This decrease, however, is thought to be a numerical artefact: due to the coarse discretization of the fracture zones, the volume of the model representing fracture intersections is too large, especially for fractures which are subparallel, the fractures are too connected. To compensate this artificial connectivity the hydraulic conductivity of the FIZ grid cells is decreased considerably. This decrease should be interpreted as a mechanism of the algorithm to compensate for the enhanced connectivity and no physical explanation should be sought. In spite of the fact that the average fracture plane hydraulic conductivities hardly change, the variances of the average fracture plane hydraulic conductivity increase during the conditioning to the transient tests. With the variance of the average fracture plane hydraulic conductivity we refer to the variations in the average conductivity for a certain fracture plane from one realisation to another. This increase is in addition to the increase observed during the conditioning to steady-state head data. Also an additional increase is found in the conductivity variance within each of the fracture planes. However, this increase is limited to the fracture planes that are affected by the conditioning to the transient head data. Figure 8 shows the conductivity perturbation applied to condition to transient head data, for a certain realisation and for one of the fracture planes. It is clear that, locally, important hydraulic conductivity changes are induced. Some parts of the fracture planes become much less conductive (with a conductivity similar to the background material), while other parts of the fracture plane have very high conductivities. Also the spatial correlation of the applied perturbations can be seen clearly in the figure; the spatial correlation is induced by the kriging interpolation. Figure 9 gives a histogram with the hydraulic conductivities for the fracture plane 20 before the inverse conditioning (just conditioned to hydraulic conductivity data) and after the inverse conditioning to the transient head data. The graphs illustrate the variance increase for the fracture plane and also show that the distributions, initially close to Gaussian, become more skewed. These tendencies were observed in all realisations, for many of the fracture planes affected by the transient tests and were already observed during the conditioning to steady-state head data and reinforced during the conditioning to transient head data. Summary on simulation outcomes In total eight realisations were conditioned to geology (structural model with fracture plane definitions), conductivity data, steady-state head data, and head drawdown data from transient tests. The following can be concluded after the complete conditioning to the experimental information: The reproduction of the piezometric heads (steady state and transient) is good for the eight realisations. The geological model with the position of 21 main fracture planes is plausible. Although locally the conductivity decreased very significantly, the average conductivities for the fracture planes were in all cases above the average background conductivity. The contrast between the average background conductivity and the average fracture plane conductivity hardly changed during the conditioning (it showed a very small increase). The contrast in the ensemble average conductivities of the different fracture planes, originally obtained from the analysis of the few data available,
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Fig. 8a–d. Hydraulic conductivity perturbation (for one single realisation) due to conditioning to hydraulic head responses from six interference tests. a 550 m depth, b 510 m depth, c 470 m depth, d fracture plane 20
decreased slightly. However, for some fracture planes, it was evident that the initial conductivity estimates were either too high or too low. More precisely, for three fracture planes their average conductivity (computed over all realisations) changed more than 0.5 log10 m/s after conditioning (the maximum change was 0.69 log10 m/s). A consistent decrease for the conductivities at the FIZ grid cells is observed. This is attributed to a numerical correction for the artificially enhanced fracture network conductivity related to the coarse discretization of the domain used. The prior estimates of the fracture plane conductivity variances were too low. In most of the fracture planes, a variance increase is observed. This variance increase is observed throughout all the realisations. Also the ensemble variances of the average fracture plane conductivities increase. Furthermore, the statistical distributions of the log conductivities in many fracture planes become more skewed (initially the distribution of the log conductivities is Gaussian). Consistent estimates of the single storativity are found. The average calibrated storativity is 2.07 10)8 m)1 and the values just range between 1.3 10)8 and 3.6 10)8 m)1 over the eight realisations.
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Fig. 9a, b. Statistical distribution of hydraulic conductivities for one single realisation. a Before conditioning to any head data, b after conditioning to steady and transient head data
The final calibrated prescribed boundary head are smaller than their initial guesses, with some pronounced decrease near the tunnel. This decrease appears consistent with the observed reduction in background piezometric head in the entire laboratory since its construction. Simulation outcomes for other conditioning rounds The results from the last conditioning round, as presented before, are thought to be the most valuable, as the realisations are the most data charged. However, it is also interesting to compare the results with earlier simulation rounds, in which less experimental information was used. In the before-last simulation round less conductivity data were used (just 101 instead of 270 conductivity data corresponding to the background, but the same
Table 3. Average fracture plane conductivities after different conditioning stages in the before-last simulation round. Compare the results with the results obtained in the last simulation round (Table 1). The averages are calculated over eight realisations after conditioning to geology and conductivity data and after conditioning to steady-state head data. The averages are calculated over seven realisations after conditioning to five short-term transient tests
Background Fr. Plane 1 Fr. Plane 2 Fr. Plane 3 Fr. Plane 4 Fr. Plane 5 Fr. Plane 6 Fr. Plane 7 Fr. Plane 8 Fr. Plane 10 Fr. Plane 12 Fr. Plane 13 Fr. Plane 15 Fr. Plane 16 Fr. Plane 17 Fr. Plane 18 Fr. Plane 19 Fr. Plane 20 Fr. Plane 21 Fr. Plane 22 Fr. Plane 23 Fr. Plane 24 FIZ
Conditioned to geology and K data
Conditioned to steady heads
Conditioned to five short-term tests
)10.20 )6.54 )6.31 )6.61 )7.28 )4.67 )7.97 )5.63 )5.43 )7.03 )7.76 )8.15 )7.68 )5.35 )6.00 )7.03 )6.70 )6.97 )8.15 )7.20 )7.04 )6.33 )6.65
)10.43 )6.55 )6.21 )6.61 )7.18 )5.40 )7.65 )5.81 )5.67 )6.99 )7.86 )8.21 )7.85 )5.91 )6.09 )7.35 )6.68 )6.51 )7.89 )6.94 )7.17 )6.35 )7.02
)10.41 )6.56 )6.17 )6.47 )7.17 )5.49 )7.69 )5.69 )5.56 )6.97 )7.87 )8.08 )7.81 )5.90 )6.06 )7.38 )6.56 )6.43 )7.14 )6.95 )7.18 )6.34 )6.94
amount of conductivity data for the fracture planes), less steady-state head data (45 instead of 51) and also less monitoring locations during transient tests (13 instead of 29). Furthermore, the geological model was slightly different (the position of fracture number 13). The average fracture plane conductivities for the seed realisations hardly differ between the before-last conditioning round and the last conditioning round. See Table 3 for the ensemble averaged average fracture plane conductivities. The difference in average fracture plane conductivity (averaged over 8 realisations in case of the before–last conditioning round) between the two simulation rounds is never larger than 0.1 log10 units. The differences remain small during the conditioning to steady-state head data. The fractures that showed a stronger conductivity increase or decrease in the last simulation round, showed a similar behaviour in the before–last simulation round. However, for two fracture planes the differences are larger. The ensemble averaged average conductivity of fracture plane 13 was )8.2 log10 (m/s) in the before–last simulation round while it was )7.4 log10 (m/s) in the last simulation round. It is interesting to notice that the most important difference is found for the only fracture plane that had not the same position in the two simulation rounds. It is also interesting to notice that the conductivity is more elevated for the case that it is supposed that the fracture plane is located on the right position. Also the
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ensemble averaged average conductivity of fracture plane 6 differs significantly between the two simulation rounds: it was )7.7 log10 (m/s) in the before–last conditioning round and )8.3 log10 (m/s) in the last conditioning round. The fracture planes 6 and 13 have the same orientation and are located close to each other. The incorrect location of fracture plane 13 in the before–last simulation round may have caused fracture plane 6 to have a higher conductivity than in the last simulation round; fracture plane 6 ‘‘compensated’’ some of the missing conductivity due to the incorrect location of fracture plane 13. Finally, a significant difference in the ensemble averaged average conductivities is found for the FIZ grid cells. Also the transient tests do not yield significant differences in average fracture plane conductivities between the before–last and last simulation round, apart from the mentioned differences. This indicates that with respect to the ensemble averaged average fracture plane conductivities the main differences can be attributed to modifications in the geological model, while the amount of head and conductivity data do not have a significant influence on this characterisation measure. Another comparison measure is the ensemble variance of the average fracture plane conductivities. Although the before–last and last simulation round yield very similar ensemble averaged average fracture plane conductivities (for almost all fracture planes), the ensemble variance of these average fracture plane conductivities differ. The variances are larger in the before–last simulation round than in the last simulation round. In any case, it should be stressed that these ensemble variances are just calculated over seven realisations in case of the before–last simulation round and over ten realisations in the last simulation round. Not enough for a reliable estimate of these statistics. The larger variance in the before– last simulation round is due to considerable higher uncertainty on the average fracture plane conductivity of the fracture planes 20, and especially, 21 and 22. In this context it is interesting to notice that in the last simulation round the additional piezometric head data were located close to the fracture planes 20, 21 and 22. It could be argued that this extra experimental information reduced especially the uncertainty on these average fracture plane conductivities. Therefore the ensemble variance of the average fracture plane conductivities is higher in the last simulation round than in the before last simulation round. However, as stated before, the estimates of the variances of the average fracture plane conductivities are based on a limited amount of realisations.
10 Discussion and conclusions Besides the findings specific to the TRUE Block Scale experiment discussed in the previous section, a more general discussion and conclusion are given next. It is possible to generate hydraulic conductivity realisations over domains of more than 54,000 grid cells conditioned to a large amount of experimental information, including not only data on conductivity but also on piezometric heads. However, at the same time it is clear that the required CPU time is the most important limitation in this study. It would have been desirable to generate more than ten realisations over a finer discretized study domain, in order to have enough realisations to start a minimum statistical analysis of the results and to have a better representation of the large number of fractures included in the domain. However, although computers will be faster and will allow for more realisations and more grid cells, another important limitation in the simulation of groundwater flow in fractured media is the numerical solution of the groundwater flow equation. Given the very large contrast in conductivity between fracture
planes and background, better solvers for large nearly singular linear system of equations have to be developed. Most probably this is a more severe limitation than the limitation on CPU time. The study illustrates how a stochastic continuum approach is able to solve in a satisfying way groundwater flow at a fractured site. The approach in which multiple statistical populations are used to explicitly incorporate information on fracture plane positions and properties, is shown to be successful. (This approach has already been successfully used in fractured media in which the fractures are defined stochastically, see Hendricks Franssen et al., 1999b; Go´ mez-Herna´ ndez et al., 2000.) At the moment, the method is able to calculate conductivity perturbations for each statistical population separately. A very interesting extension of this approach would allow the calibration of the locations of each population, that is, during the calibration process, not only the conductivity values could change but also the classification of a cell as belonging to a given population. In this particular case, a major challenge would be to develop a methodology that includes the possibility that a grid cell changes from fracture plane to background or the other way around during the conditioning process. The sequential incorporation of information is in general advantageous as compared to the simultaneous incorporation of different sources of information. On one hand, the simultaneous incorporation of all sources of information may yield instabilities; convergence is improved with the sequential addition of information. On the other hand, the sequential incorporation of conductivity data, steady-state head data and transient head data from different tests also allows a better study of the effect that the different pieces of information have on the hydraulic conductivity estimates.
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